There are two airports in Gothenburg: Gothenburg City Airport and Gothenburg Landvetter Airport. To get to the city from either you can get on the Flygbussarna (airport bus).
The stop in the center of the city at the Central Station is called Nils Ericson Terminal. From here you can get trains, trams and buses to the surrounding area or if you are staying in the city center you could walk from here.
Living in Gothenburg
If you are hunting for a place to stay, try Blocket. This is the Swedish equivalent of Gumtree.
Choose the area e.g. Göteborg and then from the 'all kategorier' box select 'lägenheter' (apartments) which can be found under 'Bostad' (Housing). Next click the radio button 'Uthyres' and then a list of available apartments to rent will be displayed. Housing is described in square meters. The flats are usually described in Swedish but if you use a translator you can get the general gist.
I have installed Ubiquity on my computer so I only have to highligh the text and it translates it for me. Use Oanda to do currency conversions.
Housing is hard to find in Sweden although it is easier during the summer months. Rent is controlled so finding a 'first-hand' contract for an apartment in a city center is virtually impossible. It is possible to sublet and that is usually what you find on Blocket. It's possible to buy in Sweden but often there are strict rules about renting out your property so you can only buy if you are going to live in the apartment.
Places you may need to go
To register to stay in Sweden for more than three months you need to go to Migrationsverket
To get a personal number after you've applied to stay in Sweden you need to go to the one of the Skatteverket offices.
If you need help finding a job then try going to the unemployment office called the Arbetsformedlingen. There are often several in a big city like Gothenburg.
If you need to get a certified copy of your passport - go to the Notarius publicus at Drottninggatan 38, 411 07 Göteborg, Tel 031-799-10 90. Open 10-12 13-15.30. The day before a helgdag (public holiday) they are only open 10-13.00. They are shut at the weekend. A certified copy of your passport takes a day or two to get back and costs 150 kroner.
All immigrants are entitled to free Swedish lessons. To register for them in Gothenburg go to Rosenlundsgatan 8 to the SFI office. You will need a personal number before you can do this.
Other useful links
The Local provides Swedish news in English as well as noticeboards and discussion boards for English speakers.